Presumably the bill would be heavily amended. Haridopolos said he favors a harder line on some issues thatn Sen. Anitere Flores, the Miami Republican who shepherded the bill before Haridopolos handed it off to Sen. J.D. Alexander, a Lake Wales Republican, on Wednesday.

Haridopolos also said state law enforcement officials should not be able to check the immigration status of people who get pulled over in a traffic stop.

Alexander later weighed in, too.

"I'm not comfortable going with anything that may be perceived as profiling," he said, adding that the Senate is considering checking a person's immigration status after he or she is read his or her rights. That, Alexander said, could help in "dramatically reducing the potential for profiling."

He said that it would have to be used on everyone: "You'd probably have to check everyone so nobody is selectively considered," he said.